On Landis’ conjecture in the plane
Journal
Comm PDE
Journal Volume
Vol 40 (2015)
Pages
766-789
Date Issued
2015
Date
2015
Author(s)
Abstract
In this paper we prove a quantitative form of Landis’ conjecture in the plane. Precisely, let W(z) be a measurable real vector-valued function and V(z) ≥0 be a real measurable scalar function, satisfying ‖W‖ L ∞(R 2) ≤ 1 and ‖V‖ L ∞(R 2) ≤ 1. Let u be a real solution of Δu − ∇(Wu) − Vu = 0 in R 2. Assume that u(0) = 1 and |u(z)| ≤exp (C 0|z|). Then u satisfies inf |z 0| =R sup |z−z 0| <1|u(z)| ≥exp (−CRlog R), where C depends on C 0. In addition to the case of the whole plane, we also establish a quantitative form of Landis’ conjecture defined in an exterior domain.
Type
journal article
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